Computing zeta functions of Kummer curves via multiplicative characters (Q1405724)

Let \(p\) be a prime number, \(a\) a positive integer, \(q=p^a\). Let \(\bar{f}\in {\mathbb F}_q[X]\) be a squarefree polynomial of degree \(d\) with \(\bar{f}(0)\neq 0\). Let \(m\) be a divisor of \(p-1\), prime to \(d\). Let \(C_{\bar{f}}\) be the Kummer curve \(Y^m=\bar{f}(X)\) with the projective smooth model \(\widetilde{C}_{\bar{f}}\). The author proves: The zeta function of \(\widetilde{C}_{\bar{f}}\) may be computed deterministically in \(\widetilde O(pa^3d^4m^3)\) bit operations. The \( \widetilde{O}\) notation means that logarithmic factors are ignored.